HARA utility maximization in a Markov-switching bond–stock market

Abstract

We present a flexible multidimensional bond–stock model incorporating regime switching, a stochastic short rate and further stochastic factors, such as stochastic asset covariance. In this framework we consider an investor whose risk preferences are characterized by the hyperbolic absolute risk-aversion utility function and solve the problem of optimizing the expected utility from her terminal wealth. For the optimal portfolio we obtain a constant-proportion portfolio insurance-type strategy with a Markov-switching stochastic multiplier and prove that it assures a lower bound on the terminal wealth. Explicit and easy-to-use verification theorems are proven. Furthermore, we apply the results to a specific model. We estimate the model parameters and test the performance of the derived optimal strategy using real data. The influence of the investor’s risk preferences and the model parameters on the portfolio is studied in detail. A comparison to the results with the power utility function is also provided.

Keywords:

• Portfolio optimization
• Markov chains
• HARA utility
• Hamilton–Jacobi–Bellman equations

JEL Classification:

• G11
• C61

Acknowledgements

We would like to thank two anonymous referees for their helpful and insightful comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

1 See Jacod and Shiryaev (2003) and Schönbucher (2003) for Girsanov’s Theorem about change of measure with jump processes. Note that the predictability required there can be achieved by considering instead of . This would not change the density process for the measure change as only integrals w.r.t. continuous measures are involved.

2 If we assume the Vasicek model for and an affine function for : , then for the short rate we again obtain a Vasicek process. As it does not increase the flexibility of the model, we define in order to avoid unnecessary complication of the notation.

3 Note that in the model we consider in section 4 we explicitly derive this solution.

4 For a definition of the differential semimartingale characteristics consult Kallsen and Muhle-Karbe (2010a).

5 We would like to thank Laslo Bollmann and Andreas Lichtenstern for coding parts of a previous version of the estimation procedure under the close guidance of Daniela Neykova.

6 For an introduction to the Baum–Welch algorithm we refer to Baum et al. (1970) and Zucchini and MacDonald (2009).

7 An introduction to the Kalman Filter can be found in Kalman (1960), Kalman and Buchy (1961) and Harvey (1989).

Additional information

Funding

Daniela Neykova gratefully acknowledges financial support from the program Women in Math Science at the Technical University of Munich.